Simplify the following expression: $y = \dfrac{4x^2- 5x+1}{x - 1}$
Solution: First use factoring by grouping to factor the expression in the numerator. This expression is in the form ${A}x^2 + {B}x + {C}$ First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(4)}{(1)} &=& 4 \\ {a} + {b} &=& &=& {-5} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $4$ and add them together. The factors that add up to ${-5}$ will be your ${a}$ and ${b}$ When ${a}$ is ${-1}$ and ${b}$ is ${-4}$ $ \begin{eqnarray} {ab} &=& ({-1})({-4}) &=& 4 \\ {a} + {b} &=& {-1} + {-4} &=& -5 \end{eqnarray} $ Next, rewrite the expression as $({A}x^2 + {a}x) + ({b}x + {C})$ $ ({4}x^2 {-1}x) + ({-4}x +{1}) $ Factor out the common factors: $ x(4x - 1) - 1(4x - 1)$ Now factor out $(4x - 1)$ $ (4x - 1)(x - 1)$ The original expression can therefore be written: $ \dfrac{(4x - 1)(x - 1)}{x - 1}$ We are dividing by $x - 1$ , so $x - 1 \neq 0$ Therefore, $x \neq 1$ This leaves us with $4x - 1; x \neq 1$.